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# Structural Design for Cold Region Engineering - PowerPoint PPT Presentation

Structural Design for Cold Region Engineering. Lecture 14 Thory of Plates Shunji Kanie. Theory of Plates Kirchhoff Plate. Kirchhoff Plate. Pure Bending. Such as Bernoulli Euler. Assumptions. Isotropic and homogeneous The thickness of the plate is thin

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### Structural Design forCold Region Engineering

Lecture 14 Thory of Plates

Shunji Kanie

Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Pure Bending

Such as Bernoulli Euler

Assumptions

Isotropic and homogeneous

The thickness of the plate is thin

(Comparatively to the length and width )

Linear filaments of the plate

(Even after the deformation)

Kirchhoff hypothesis

Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Length : a in x direction

Width : b in y direction

Thickness : h in z direction

Deflection

Rotation angle

Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Rotation angle

Displacement due to deflection

Theory of PlatesKirchhoff Plate

Kirchhoff Plate

Stress-Strain Relation

Plane Stress !

Theory of PlatesKirchhoff Plate

Sectional Force

Bending Moments and Torsional Moment are calculated at least

Theory of PlatesKirchhoff Plate

X direction

y direction

z direction

Theory of PlatesKirchhoff Plate

z direction

Theory of PlatesKirchhoff Plate

Governing Equation

Theory of PlatesKirchhoff Plate

Governing Equation

Theory of PlatesKirchhoff Plate

Introducing Laplacian

Theory of PlatesKirchhoff Plate

Boundary Conditions

Simple support

Fixed support

Free support

Effective transverse shear

Kirchhoff force

Theory of PlatesSolution

Simply Supported Plate

Assuming Deformation

Boundary Condition?

Theory of PlatesSolution

Simply Supported Plate

Governing Equation

Assumed Deformation

Theory of PlatesSolution

Simply Supported Plate

Assumed Deformation

Deformation

Theory of PlatesSolution

Simply Supported Plate

Deformation

Bending & Twisting Moments

Theory of PlatesSolution

Simply Supported Plate

Bending & Twisting Moments

If we are very LUCKY enough

Theory of PlatesSolution

Simply Supported Plate

If

qmn is successfully calculated and we can have the solution

Is there any good idea if q is uniformly distributed load?

Theory of PlatesSolution

Simply Supported Plate

Apply Double Fourier Expansion for q

Theory of PlatesSolution

Simply Supported Plate

When q is constant as q0

You can solve the problem for any shape of load distribution

Theory of PlatesSolution

Plate supported like

Assuming Deformation

Single Fourier Expansion

Theory of PlatesSolution

Plate supported like

If q is constant in x direction

m=1,3,5,…….

Theory of PlatesSolution

Plate supported like

If q is linear in x direction

m=1,2,3,4,…….

Theory of PlatesSolution

Plate supported like

If q is constant in x direction

m=1,3,5,…….

If q is linear in x direction

m=1,2,3,4,…….

Solve

Theory of PlatesSolution

Plate supported like

Solve

General Solution

Characteristic Equation

Theory of PlatesSolution

Plate supported like

General Solution

Singular Solution

should be constant such as

Theory of PlatesSolution

Plate supported like

General Solution

Singular Solution

Solution

Theory of PlatesSolution

Difference Method

Theory of PlatesSolution

Difference Method

Governing Equation

Simple support

Fixed support

Theory of PlatesSolution

Galerkin Method

Assume Approximation

Governing Equation

Weighted Residual

Same with Double Fourier